Approximations and reanalysis over a parameter interval for dynamic design

Abstract

In many design search and optimization situations, the objective of optimization or design improvement is closely related to one or more natural frequencies of a dynamic system. The problem typically involves solving an eigenvalue problem repeatedly for a large set of values of the parameters which describe the system or the structure being designed. When the system is complex, the parameter space tends to be large (i.e, the number of parameters that define the geometry, material properties, etc is large). The situation is further complicated by the fact that complex geometries usually require a large number of degrees of freedom for a reasonably accurate analysis (e.g. an ap9propriate finite-element analysis). For this reason, the design search and optimization problem tends to be computationally very demanding. In most situations, it is this step involving the solution of the eigenproblem associated with the free vibration that consumes most computational resources.
The present note is motivated by this engineering need. A method based on interpolation for an approximate estimation of eigenvalues is presented. Instead of the usual approximation around a reference design, the approach here is to find approximations (at possibly several points) over an interval of the parameter of interest. This problem has been attempted via an alternative route by Bhaskar (1) where the approximations are sought for eigenvectors by interpolating the mode shapes themselves over the parameter interval of interest; trial vectors obtained in this manner were used in a Rayleigh-quotient approximation. The present scheme differs in that it provides approximations for eigenvalues directly form the estimates that use exact eigensolutions at the terminal points of the design interval.

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